Corollary 7.1.7.14. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories, let $B$ be a simplicial set, and let $F': \operatorname{Fun}(B,\operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}(B, \operatorname{\mathcal{D}})$ be given by composition with $F$. Let $\overline{f}: K^{\triangleright } \rightarrow \operatorname{Fun}(B, \operatorname{\mathcal{C}})$ be a diagram. Assume that, for every vertex $b \in B$, the composition
\[ K^{\triangleright } \rightarrow \operatorname{Fun}(B, \operatorname{\mathcal{C}}) \xrightarrow { \operatorname{ev}_ b } \operatorname{\mathcal{C}} \]
is an $F$-colimit diagram in the $\infty $-category $\operatorname{\mathcal{C}}$. Then $\overline{f}$ is an $F'$-colimit diagram in the $\infty $-category $\operatorname{Fun}(B, \operatorname{\mathcal{C}})$.